By elimination of a,b,cfrom the homogeneous equations $$x=ab$$−c, $$y=bc$$−a, $$z=ca$$−bwhere a,b,cnot all zero.

Correct option is 2xy+yz+zx=−1

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By elimination of $a, b, c$ from the homogeneous equations $x = \frac{a}{b - c}, y = \frac{b}{c - a}, z = \frac{c}{a - b}$
where $a, b, c$ not all zero.

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xy+yz+zx=1

xy+yz+zx=−1

x+y+z=0

x+y+z=1

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detailed solution

Correct option is B

a−xb+xc=0→(1) ya+b−yc=0→(2)za−zb−c=0→(3)Since a, b, c are not all zero, the above system has a non-trivial solution.By eliminating a,b,cfrom (1),(2),(3),we get1-xxy1-yz-z-1=0 ⇒xy+yz+zx=−1

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By elimination of a,b,cfrom the homogeneous equations x=ab−c, y=bc−a, z=ca−bwhere a,b,cnot all zero.

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By elimination of $a, b, c$ from the homogeneous equations $x = \frac{a}{b - c}, y = \frac{b}{c - a}, z = \frac{c}{a - b}$
where $a, b, c$ not all zero.